snap to the ground/grid
If I understand what you mean, you wish the bottom of your sphere object to touch exactly the grid, as if it "felt" on a virtual ground (that doesn't exist).

To achieve this, you need to tell blender exactly what you mean by bottom, unfortunately. I.e., you need to (temporarily) move the origin of your sphere to the object's mesh feature you wish blender snaps everywhere, even to the grid.

E.g. you need to go into edit mode, select only the bottom (vertex?) of your mesh,



then go back to object mode, and select "Object > Set Origin > Origin to 3D cursor"



After doing so, the snap command referred to the absolute grid snap 



will allow you to exactly snap the (temporary) origin of your object as you wish.



Then you can restore the origin of your object using another option from the same menu above, i.e. "Origin to Geometry"

If you had a "floor" object like a plane or else, you could maybe use a rigid body simulation to let your sphere fell onto the "floor", but that requires much more setup for a simple task like this, although it could be useful though to let  as series of objects (e.g. tree shaped objects) to fell onto another one (e.g. a mountain shaped big object) instead of placing them manually...

The rigid body simulation will also be quite real, and your object would need to sit perfectly on the floor or, as maybe in the ICOsphere above, it would roll on a side, etc. Not recommended in this case.

I hope this answers you.